Torelli theorem for the moduli space of symplectic parabolic Higgs bundles

TL;DR

This paper proves a Torelli-type theorem for the moduli space of stable symplectic parabolic Higgs bundles, showing that the moduli space uniquely determines the underlying Riemann surface and marked points.

Contribution

It establishes that an isomorphism between such moduli spaces implies an isomorphism of the underlying marked Riemann surfaces.

Findings

Moduli space determines the underlying Riemann surface.

Isomorphism of moduli spaces implies surface isomorphism.

Results extend Torelli theorems to parabolic Higgs bundles.

Abstract

Let $(X,D)$ and $(X',D')$ be two compact Riemann surfaces of genus $g \geq 4$ with the set of marked points $D \subset X$ and $D' \subset X'$. Fix a parabolic line bundle $L$ with trivial parabolic structure. Let $\mathcal{N}_{\textnormal{Sp}}(2m,\alpha,L)$ and $\mathcal{N}'_{\textnormal{Sp}}(2m,\alpha,L)$ be the moduli spaces of stable symplectic parabolic Higgs bundles over $X$ and $X'$ respectively, with rank $2m$ and fixed parabolic structure $\alpha$, with the symplectic form taking values in $L$. We prove that if $\mathcal{N}_{\textnormal{Sp}}(2m,\alpha,L)$ is isomorphic to $\mathcal{N}'_{\textnormal{Sp}}(2m,\alpha,L)$, then there exist an isomorphism between $X$ and $X'$ sending $D$ to $D'$.